# Applied Mathematics/Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering. Laplace transform is denoted as $\displaystyle {\mathcal {L}}\left\{f(t)\right\}$ .

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

## Definition

For a function f(t), using Napier's constant"e" and complex number "s", the Laplace transform F(s) is defined as follow:

$F(s)={\mathcal {L}}\left\{f(t)\right\}(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$

The parameter s is a complex number:

$s=\sigma +i\omega ,\,$  with real numbers σ and ω.

This $F(s)$  is the Laplace transform of f(t).

## Examples of Laplace transform

Examples of Laplace transform
function result of Laplace transform
$C$  (constant) ${\frac {C}{s}}$
$t$  ${\frac {1}{s^{2}}}$
$t^{n}$  (n is natural number) ${\frac {n!}{s^{n+1}}}$
${\frac {t^{n-1}}{(n-1)!}}$  ${\frac {1}{s^{n}}}$
$e^{at}$  ${\frac {1}{s-a}}$
$e^{-at}$  ${\frac {1}{s+a}}$
${\rm {cos}}\ \omega t$  ${\frac {s}{s^{2}+{\omega }^{2}}}$
${\rm {sin}}\ \omega t$  ${\frac {\omega }{s^{2}+{\omega }^{2}}}$
${\frac {t^{n-1}}{\Gamma (n)}}$  ${\frac {1}{s^{n}}}$  (n>0)
$\delta (t-a)$  (Delta function) $e^{-as}$
$H(t-a)$  (Heaviside function) ${\frac {e^{-as}}{s}}$

## Examples of calculation

(1)Suppose $f(t)=C$  (C = constant)
$\int _{0}^{\infty }e^{-st}C\,dt$
$={\frac {C}{s}}$
$=F(s)$

(2)Suppose $f(t)=e^{-at}$
$\int _{0}^{\infty }e^{-st}\cdot e^{-at}\,dt$
$=\int _{0}^{\infty }e^{-(s+a)t}\,dt$
$=\left[{\frac {-e^{-(s+a)t}}{s+a}}\right]_{0}^{\infty }$
$={\frac {1}{s+a}}$